I've tried unscrewing the part outlined in red below. No dice.
Do I just need to remove the whole rod (above the red outline) and replace it?)
If so, can anyone point me to instructions for them. I can't see in there. Worried that I'll pull it out and not be able to put it back in.

(The deeper I go in the plumbing the more things I find that there is no modern fitting-part :)

My new assembly replaces the parts outlined in red. the new assembly has a female thread. I am HOPING that the thicker part of the pipe (just above the black "collar" stopper) is screwed onto the pipe above.

However, I cannot unscrew it. There is nothing to get a hold of. It's perfectly round.
I've tried dw40, and let it sit for 20 minutes. No love.

I'll try wacking it with a board in the direction along which it moves in when it screws (perpendicular to the threads per [this answer][2])

I've never seen a joint like that before (but I'm not a professional plumber either!). Can you take a few more pictures at a different angle? I'm not even certain which end is the supply end. You mention "Above the red" I'm assuming you mean the upper part of the picture? Assuming that is the supply side, is it sweated on the other end? This looks like it is some kind of quick disconnect; like the white portion is press fit on horizontally (respective to the photo). The black part is a bushing? And the 1/2" OD to 5/8" OD is a tapered bushing? Probably tubing cut it and sweat on a threaded end
– noybmanSep 8 '17 at 22:13

The photo is taken from above. The water supply is from the top of the photo. The faucet handle is at the bottom. And yes, the white part is a new part I connected but it is not size-compatible with the old part. So I have to replace the part outlined in red as well. But it's ATTACHED to the Stem, So I have to replace that as well.
– Clay NicholsSep 10 '17 at 1:28

1 Answer
1

That part is called the stem, and is typically fully replaced, or more often, you simply replace the o rings and, well, as noybman mentioned, the bushing(s).

I have a feeling, if you pull the stem completely out, you'll fine the little brass part you're really looking for. There's a base at the back of that stem that closes the valve, and it will be replaceable on some spigots, in whole or in part. If yours is replaceable, it is typically adjustable as well.

Do a google image search for your spigot under 'repair kit'. You'll get a very handy diagram.